Multiple counters automata, safety analysis and presburger arithmetic

  • Hubert Comon
  • Yan Jurski
Regular Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1427)


We consider automata with counters whose values are updated according to signals sent by the environment. A transition can be fired only if the values of the counters satisfy some guards (the guards of the transition). We consider guards of the form yi#yj + ci,j where yi is either xí or xi, the values of the counter i respectively after and before the transition, and # is any relational symbol in {=,≤,≥,>,<}. We show that the set of possible counter values which can be reached after any number of iterations of a loop is definable in the additive theory of ℕ (or ℤ or ℝ depending on the type of the counters). This result can be used for the safety analysis of multiple counters automata.


Model Check Finite State Automaton Additive Theory Presburger Arithmetic Counter Automaton 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. 1.
    R. Alur and D. Dill. Automata for modeling real-time systems. In Proc. 17th Int. Coll. on Automata, Languages and Programming, Warwick, LNCS 443, pages 322–335. Springer-Verlag, 1990.Google Scholar
  2. 2.
    O. Bernholtz, M. Vardi, and P. Wolper.An automata-theoretic approach to branching time model checking. In Proc. Computer Aided Verification, 1994.Google Scholar
  3. 3.
    B. Boigelot. Linear operators and regular languages (ii). Unpublished draft, jan 1997.Google Scholar
  4. 4.
    B. Boigelot and P. Wolper. Symbolic verification with periodic sets. In Computer Aided Verification, Proc. 6th Int. Conerence, LNCS, Stanford, June 1994. Springer-Verlag.Google Scholar
  5. 5.
    T. Bultan, R. Gerber, and W. Pugh. Symbolic model checking of infinite state systems using presburger arithmetic. In O. Grumberg, editor, Proc. Computer Aided Verification, volume 1254 of LNCS, Haifa, Israel, 1997. Springer-Verlag.Google Scholar
  6. 6.
    T. Cormen, C. Leiserson, and R. Rivest. Introduction to algorithms. MIT Press, 1990.Google Scholar
  7. 7.
    P. Cousot and N. Halbwachs. Automatic discovery of linear restraints among variables of a program. In Proc. Int. Conf. on Princinples Of Programming Languages (POPL), 1978.Google Scholar
  8. 8.
    J. Esparza. Decidabihty of model checking for infinite-state concurrent systems. Acta Informatica, 34:85–107, 1997.CrossRefGoogle Scholar
  9. 9.
    L. Fribourg. A closed form evaluation for extending timed automata. Technical Report 1998-02, Laboratoire Spécification et Vérification, ENS Cachan, Mar. 1998.Google Scholar
  10. 10.
    L. Fribourg and H. Olsen. A decompositional approach for computing least fixedpoint of datalog programs with z-counters. J. Constraints, 1997.Google Scholar
  11. 11.
    L. Fribourg and J. Richardson. Symbolic verification with gap-order constraints. Research Report LIENS-96-3, Ecole Normale Supérieure, Paris, Feb. 1996.Google Scholar
  12. 12.
    N. Halbwachs. Delay analysis in synchronous programs. In Proc. Computer Aided Verification, LNCS 697, pages 333–346. Springer-Verlag, 1993.Google Scholar
  13. 13.
    M. Minsky. Computation, Finite and Infinite Machines. Prentice Hall, 1967.Google Scholar
  14. 14.
    R. Parikh. On context-free languages. J. ACM, 13, 1966.Google Scholar
  15. 15.
    P. Revesz. A closed form for datalog queries with integer order. In Proc 3rd International Conference on Database Theory, pages 187–201, Paris, 1990.Google Scholar
  16. 16.
    W. Thomas. Automata on infinite objects. In J. van Leeuwen, editor, Handbook of Theoretical Computer Science, pages 134–191. Elsevier, 1990.Google Scholar
  17. 17.
    D. Toman, J. Chomicki, and D. S. Rogers. Datalog with integer periodicity constraints. In Int. Symp. on Logic Programming, 1994.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Hubert Comon
    • 1
  • Yan Jurski
    • 1
  1. 1.LSV, ENS CachanCachan cedexFrance

Personalised recommendations